Power contamination and domination on the grid

The contamination game of a grid graph G(n,m) is a dynamic variant of the domination, similar to the power domination. This standard is introduced by Haynes, Hedetniemi and Henning in 2002, which is initially defined as a basic domination for a set of vertices S in a graph G, and then a propagation of this domination in all vertices of G, while starting with S. On the other hand, the contamination phenomena in G(n,m) is interpreted by an evolutionary automaton cellular, which aims to propagate viruses according to a given propagation rules. In this paper, we define a mathematical self-playing game called a contamination game based on the power domination, in which, we identify the minimum number of contaminant cells for G(n,m), called the contamination number and denoted γ(G(n,m)).